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My car also has FSD but my account page does not... some times the account pages are inaccurate, at least in my experience. I'm not sure what "wins" when the car gets sold though, would they turn it off?
Sounds to me like a bad idea for you to buy this particular automobile. You already are having issues with it and you haven't purchased it yet. I would wait until Tesla offers used Model 3's and have them be responsible for all of these details. That used car dealer should know everything about the car if they want to sell it. They don't even know what to ask the original owner about the car.
If you have EAP, the cost to upgrade after the fact to FSD is $5K, at least it is for my husband's car. He's an original owner. Not sure if that makes a difference.
Welcome to the family. Check out our Kansas City Tesla Enthusiasts Club. Next event is at Jowler winery for Earth Day. April 27th. Members get two free tastings.